Complex images can be transformed into a more compact representation by mapping of the image using orthogonal basis functions. As an example, principal component analysis (PCA) is a mathematical procedure that transforms a number of, possibly, correlated variables into a (smaller) number of uncorrelated variables (i.e., independent, orthogonal, variables) called principal components. The first principal component accounts for as much of the variability in the data as possible, and each succeeding component accounts for as much of the remaining variability as possible. A principal component analysis can be considered as a rotation of the axes of the original variable coordinate system to new orthogonalaxes, called principal axes, such that the new axes coincide with directions of maximum variation of the original observations.
Image decomposition using orthogonal basis functions has many practical applications including image reconstruction, signal de-noising, blind source separation and data compression.
The question remains as to how many principal components (basis functions) need to be retained for accurate image reconstruction. A critical component is the appropriate choice of cutoff number N that provides the desired dimensional reduction without loss of relevant data. Many methods, both heuristic and statistically based have been proposed to determine the choice of cutoff number N.
Representative methods include: the broken-stick model, the Kaiser-Guttman test, Log-Eigenvalue diagram, cumulative percentage of total variance, Velicer's partial correlation procedure, Cattell's scree test, cross-validation, bootstrapping techniques, and Bartlett's test for equality of eigenvalues. Most of the aforementioned techniques suffer from an inherent subjectivity and Jolliffe, (2005) observes that “attempts to construct rules having more sound statistical foundations seem, at present, to offer little advantage over simpler rules in most circumstances”.
Additionally, many of these techniques provide a single measure of the accuracy of the image reconstruction, but do not provide information on the spatial structure of the reconstruction errors.